For the two-degree-of-freedom system considered in Problem 6.85, find the free-vibration response, \(x_{1}(t)\) and \(x_{2}(t)\), using the modal equations derived in Problem 6.85 for the following initial conditions: \(x_{1}(0)=2, x_{2}(0)=3, \dot{x}_{1}(0)=\dot{x}_{2}(0)=0\).

Data From Problem 6.85:-

Consider the free-vibration equations of an undamped two-degree-of-freedom system:

\[[m] \ddot{\vec{x}}+[k] \vec{x}=\overrightarrow{0}\]

with

\[[m]=\left[\begin{array}{ll}1 & 0 \\0 & 4\end{array}\right] \text { and }[k]=\left[\begin{array}{rr}8 & -2 \\-2 & 2\end{array}\right]\]

a. Find the orthonormal eigenvectors using the mass normalized stiffness matrix.

b. Determine the principal coordinates of the system and obtain the modal equations.